If you look at this topic you might notice same audio card and busID as you're running. ori0n's mysteriously started working the next day. Can we hope the same for you?

A couple of things that might improve the help here:
1. Is HDMI how you want sound? Or from your speakers/headphones? If the latter, in pavucontrol's Configuration tab is there a second Built-in Audio dropdown to select Analog Stereo Duplex? If this is accomplished, go back to the Output Devices tab and you should be able to select headphones or speakers.
2. Have you installed anything related to JACK (jackd, Cadence, Qjackctl, etc?) If so, you might need to kill jack to get pulseaudio running again with killall jackd followed by pulseaudio --start (there are other ways to get them to cooperate that I won't go into here.)

I've seen your name and assistance all over this forum as I have been searching for answers, so thank you for helping me out as well. It is truly appreciated!

OK, to answer your questions:
1. HDMI is the only option, it seems. Some context: This is an HP ENVY all-in-one desktop PC. What's REALLY interesting is there is no external headphone jack, so I have no way of testing that, either. Now, in pavucontrol and the configuration tab, NO cards are available for configuration.
2. Nothing related to JACK.

I did follow the topic you linked and installed some of the packages mentioned in that thread that were missing. I will reboot!

Also, when you have time, paste back the results of inxi -A as well as pacmd list-sinks

When pasting large blocks of text, use the code button (looks like </>) to help retain formatting. For me, when I hit that button I can rightclick anywhere in the text area to paste and it automatically pastes it between the code and /code tags.

Damn. I typed eveything below then realized I've never seen your specs. Before trying the below, please post the results of inxi -Fxz.

Here's something that may help (but I'm not really counting on it.)

You said in your first post:

As you'll see, I've upgraded the kernel to the latest stable release, no effect. Will be reverting back to 4.13.0-45.50~16.04.1

I now see you had been using a kernel not supported by Mint/Ubuntu (4.17)

Maybe try using the Update Manager to update to the latest kernel in the 4.15 series and see if that resolves the issue. After opening the Update Manager click on View > Linux kernels, then click on 4.15 on the left, select 4.15.0-23 and click on Install.

EDIT: I should have asked this earlier as well. What exactly happens when you type alsamixer?